Proving theorem

Kayhan Erciyeş writes in the 2nd Chapter of his book, "Discrete Mathematics and Graph Theory", the following about the Proving theorem:

"A mathematical system consists of axioms, definitions, theorems and various other structures. A theorem is a proposition that can be proved to be true and an argument that establishes the truth of a statement is called a proof. Proving a theorem can be accomplished by a direct method or indirectly. Proving propositions that involve quantifiers requires careful reasoning. In this chapter, we review main methods of proof which include direct and indirect methods, proving propositions with quanti- fiers, proof by cases and review general principles of proofs."

I collected the answers for the most substantial questions on the Proving theorem.

1. What is an axiom?

An axiom is a fundamental principle or statement accepted as true without proof. In mathematics and logic, axioms serve as the basis from which other truths are derived through logical reasoning.

2. Difference between a theorem, a lemma, and a corollary?

• Theorem: A statement proven based on previously established statements, such as other theorems and generally accepted truths (axioms).

• Lemma: A preliminary proposition or proof used to help prove a larger theorem. It is a "helper" theorem that plays an intermediate role in proving other statements.

• Corollary: A statement that follows readily from a previously proven statement. It is often seen as a direct consequence of a theorem.
  

3. Example of modus ponens in everyday life

"If it is raining, then I will carry an umbrella. It is raining. Therefore, I will carry an umbrella."

4. Example of modus tollens in everyday life

"If it is raining, then the ground will be wet. The ground is not wet. Therefore, it is not raining."

5. Example of disjunctive syllogism in everyday life

"I will either take a bus or walk to work today. I did not take a bus. Therefore, I walked to work today."

6. What is a trivial proof?

A trivial proof is used when the truth of a proposition is self-evident or follows directly from definitions.

Example: Proving that any number multiplied by zero is zero. This is trivial because of the definition of multiplication by zero.

7. What is a vacuous proof?

A vacuous proof establishes the truth of a statement that is true by virtue of its premise never being true.

Example: "If the moon is made of green cheese, then I am a millionaire." The statement is vacuously true since the moon is not made of green cheese.

8. Comparing contrapositive and contradiction methods of proof

• Contrapositive Proof: Shows that the contrapositive of a statement is true, thereby proving the original statement. It uses the logical equivalence of a statement and its contrapositive.

• Contradiction Proof: Assumes the statement to be false and shows that this assumption leads to a contradiction, thus the statement must be true. It does not rely on the contrapositive but on proving an assumption false through contradiction.
  

9. Difference between proving a conditional and a biconditional statement

• Conditional Statement (If P, then Q): Proving it typically involves assuming P is true and then showing Q follows.

• Biconditional Statement (P if and only if Q): Requires proving both the conditional (If P, then Q) and its converse (If Q, then P), establishing a two-way relationship.
  

10. Negation method of proof for universal and existential statements

• To prove a universal statement (All X are Y) is false, you provide a counterexample where X is not Y.

• To prove an existential statement (There exists an X that is Y) is false, you show that no such X can exist under the given conditions, often through contradiction.
  

11. When is "proof by cases method" used?

Proof by cases is used when a proposition can be divided into a finite number of cases, and the proposition is proven by separately proving it for each case. It's particularly useful when a direct proof is not feasible, but the proposition can be exhaustively examined through distinct, manageable cases.